Question 140619
{{{(x-4)/(x+3)>=2}}} Start with the given inequality



{{{(x-4)/(x+3)-2>=0}}} Subtract 2 from both sides



{{{(x-4)/(x+3)-(2)(x+3)/(x+3)>=0}}} Multiply 2 by {{{(x+3)/(x+3)}}}



{{{(x-4)/(x+3)-(2(x+3))/(x+3)>=0}}} Multiply



{{{(x-4-2(x+3))/(x+3)>=0}}} Combine the fractions



{{{(x-4-2x-6)/(x+3)>=0}}} Distribute



{{{(-x-10)/(x+3)>=0}}} Combine like terms



{{{x+3=0}}} Set the denominator equal to zero



{{{x=0-3}}}Subtract 3 from both sides



{{{x=-3}}} Combine like terms on the right side



So the vertical asymptote is {{{x=-3}}}. So one critical value is {{{x=-3}}}


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{{{-x-10=0}}} Set the numerator equal to zero



{{{-10=x}}} Add x to both sides



So another critical value is {{{x=-10}}}





So our critical values are {{{x=-3}}} and  {{{x=-10}}} 


Now set up a number line and plot the critical values on the number line


{{{number_line( 600, -15, 10,-3,-10)}}}




So let's pick some test points that are near the critical values and evaluate them.



Let's pick a test value that is less than {{{-10}}} (notice how it's to the left of the leftmost endpoint):


So let's pick {{{x=-11}}}


{{{(x-4)/(x+3)>=2}}} Start with the given inequality



{{{(-11-4)/(-11+3)>= 2}}} Plug in {{{x=-11}}}



{{{1.875>=2}}} Evaluate and simplify the left side





Since the inequality is false, this means that the interval does <b>not</b> work. So this interval is <b>not</b> in our solution set and we can ignore it.


   



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Let's pick a test value that is in between {{{-10}}} and {{{-3}}}:


So let's pick {{{x=-6}}}


{{{(x-4)/(x+3)>=2}}} Start with the given inequality



{{{(-6-4)/(-6+3)>= 2}}} Plug in {{{x=-6}}}



{{{3.3333>= 2}}} Evaluate and simplify the left side




Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.


   So part our solution in interval notation is <font size="8">[</font>*[Tex \LARGE -10,-3]<font size="8">)</font>




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Let's pick a test value that is greater than {{{-3}}} (notice how it's to the right of the rightmost endpoint):


So let's pick {{{x=-2}}}


{{{(x-4)/(x+3)>=2}}} Start with the given inequality



{{{(-2-4)/(-2+3)>= 2}}} Plug in {{{x=-2}}}



{{{-6>= 2}}} Evaluate and simplify the left side


Since the inequality is false, this means that the interval does <b>not</b> work. So this interval is <b>not</b> in our solution set and we can ignore it.


   



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Summary:


So the solution interval notation is:


 <font size="8">[</font>*[Tex \LARGE -10,-3]<font size="8">)</font>